# A numerical investigation of laminar flow over a backward facing inclined step

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## Abstract

The aim of the present study is to analyze the two dimensional flow over a backward-facing-inclined step in laminar flow regime. The inspiration for the present work is derived from the fact that in automobile industry, analyzing the flow over an inclined step shall help in understanding the characteristics of the rear vehicle wake. A considerable percentage of the energy needed to propel the vehicle is dissipated by the vorticity generated in the rear of the vehicle, hence it is of utmost importance to understand the properties of the wake. In the present paper, the flow over a backward step is initially analyzed and the results are compared with the existing literature to validate the code developed. The inclined step simulations were carried out by varying different aspects of the geometry i.e. different tilts, several upstream lengths and a range of different Reynolds numbers. Critical Reynolds numbers for vortex shedding in the wake of different step inclinations have been analyzed for all cases studied. A discussion on the time-averaged drag and lift coefficients as a function of Reynolds number and for all cases undertaken, are among the results presented. Among the conclusions, it is particularly interesting to point out that the inclination angle of 15° was found to be the critical angle for vortex shedding, after which critical Reynolds number remains constant.

## Keywords

Backward facing step Inclined step Critical Reynolds number Rolling up shear layer Boundary layer Laminar flow## List of symbols

- a
Height of the inclined step (m)

- C
_{D} Drag coefficient

- C
_{L} Lift coefficient

- F
_{D} Dimensional drag force (N)

- F
_{f} Flux through face f of the control volume

- F
_{L} Dimensional lift force (N)

- F
_{p} Dimensional normal force (N)

- F
_{s} Dimensional shear force (N)

- h
Height of the non-orthogonal physical domain

- l
Length of the non-orthogonal physical domain

- L
_{r} Non-dimensional vortex recirculation length

- \({\hat{\text{n}}}\)
Outward normal of the surface S

- P
Non-dimensional pressure

- Re
Reynolds number

- S
Surface onto the control volume

- t
Non-dimensional time

- ul
Upstream length of the physical domain

- u
Non-dimensional velocity X direction

- U
Free stream velocity in X direction at the inlet (m/s)

- v
Non-dimensional velocity Y direction

- \(\forall\)
Control volume (m

^{3})- x
Non-dimensional Eulerian coordinates in horizontal direction

- x
_{p} Dimensional horizontal coordinate of the physical domain (m)

- y
Non-dimensional Eulerian coordinates in vertical direction

- y
_{p} Dimensional vertical coordinate of the physical domain (m)

- δ
Non-dimensional boundary layer thickness on the upstream surface

- η
Non-dimensional coordinates of transformed domain in vertical direction

- θ
Angle of inclination (deg)

- ξ
Non-dimensional coordinates of transformed domain in horizontal direction

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